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June 20, 2013

The HoTT Book

Posted by Mike Shulman

Many of you have heard murmurings about this book for several months now. I’m happy to report that it’s now out!

At the link you can download electronic versions for screen, ereader, or printer, and order paperback and hardback bound copies at cost from The book is CC BY-SA licensed, which in particular means it will always be freely available.

Posted at 7:30 PM UTC | Permalink | Followups (52)

June 14, 2013

Quasicrystals and the Riemann Hypothesis

Posted by John Baez

Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively. If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank. But when Dyson says this, it’s a lot more interesting. So I’ve been trying to understand his remarks on this. And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture.

But still, I could use a lot of help: I don’t have much time for number theory, and a few pointers from experts could keep me from going down dead ends.

Posted at 9:41 PM UTC | Permalink | Followups (25)

The Selected Papers Network

Posted by John Baez

Here it is at last: the Selected Papers Network. Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers. Christopher Lee, in bioinformatics at UCLA, has developed software that makes this work. Try it out!

What’s cool about this system is that it’s federated. Instead of locking up your comments within its own website—the “walled garden” strategy followed by many other service—it explicitly shares these data in a way that people not on the Selected Papers Network can easily see. Any other service can see and use them too.

Posted at 7:40 PM UTC | Permalink | Post a Comment

June 7, 2013

G2 and the Rolling Ball

Posted by John Huerta

I just returned from a month at Hong Kong University, visiting James Fullwood, an algebraic geometer who likes to think about the mathematics of string theory. There, I gave a colloquium on G2 and the rolling ball, a paper John Baez and I wrote that is due to appear in Transactions of the AMS. This project began over a decade ago in conversations between John and Jim Dolan, later continued between Jim and me. Though Jim opted not to be a coauthor, his insights were crucial.

I would like to tell you about this paper, but I’ll warm up with a puzzle—one you’ve seen in several guises if you’ve read John Baez’s posts about this paper, but well-worth revisiting.

Puzzle: Roll a ball of unit radius on a fixed ball of radius RR, being careful not to let your ball slip or twist as you roll it. Suppose you roll along a great circle from the North Pole to the South Pole and back to the start at the North. How many 360 360^\circ turns did the ball make as it rolled?

One ball rolling on another

Here’s a variant of this puzzle you can work out very concretely: for R=1R = 1, roll one coin around another of the same kind, without slipping, and count the number of times the rolling coin turns.

Below the fold, I’ll give you the answer, and I’ll also tell you about something amazing that happens when R=3R = 3, bringing in the exceptional Lie group G 2\mathrm{G}_2 and a funky 8-dimensional number system called the split octonions.

Posted at 9:44 AM UTC | Permalink | Followups (8)

June 3, 2013

Torsors and Enriched Categories

Posted by Simon Willerton

In another instalment of my occasional series on ‘Things you didn’t realise were enriched categories (unless you’re an expert!)’ I want to talk about torsors and how they can be considered enriched categories where the enriching category is a group, considered as a discrete monoidal category.

I’ll start off by telling you what a torsor is and then explain how it can be thought of as having something like a ‘group-valued distance’ and how this relates to enriched category theory.

Posted at 12:10 AM UTC | Permalink | Followups (34)

June 1, 2013

Welcome, Emily

Posted by Tom Leinster

I’m happy to announce that Emily Riehl has joined us as a host at the Café. Emily is based at Harvard, and you can deduce some of her interests from the many guest posts she has written for us before:

Welcome, Emily!

Posted at 1:14 PM UTC | Permalink | Followups (3)